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Complexity and recurrence in Hamiltonian systems
Project responsable Félix Schlenk
   
Team member Agnes Gadbled
Yuri Chekanov
   
Abstract The most interesting dynamical systems are those describing systems without friction, and many among these can be described as Hamiltonian systems. Since there is no friction, one can expect that Hamiltonian systems are complicated. This is what we aim to prove in the first part of the project for many natural Hamiltonian systems. The origin of Hamiltonian dynamics is the study of the motion of planets, which move - fortunately! - on (almost) closed orbits. The search for closed orbits is therefore a fundamental and particularly beautiful problem in this field. In the second part of the project, we aim to show that for our class of Hamiltonian systems, every energy surface carries infinitely many closed orbits. We shall first try to prove this for a special but very natural class of Hamiltonian systems. The configuration spaces we consider are those closed manifolds whose (based or free) loop space is complicated in the sense that the dimension of its homology grows exponentially fast. Most closed manifolds belong to this class. The idea is that the fast homological growth of the configuration space Q should lead to a fast growth of the orbit complexity and of the number of closed orbits for ALL natural Hamiltonian flows over Q at each energy level. In technical terms, the first project aims to show that on rationally hyperbolic manifolds, each Reeb flow on the spherization has positive topological entropy. This would extend famous results by Dinaburg, Gromov and Paternain on geodesic flows. The second project aims to show that the number of closed orbits of these Reeb flows grows exponentially fast in time. This would extend work by Gromov on closed geodesics, and would be a strong quantitative version of the Weinstein conjecture for these systems.
   
Keywords Hamiltonian Dynamics, Lagrangian submanifolds, Lagrangian tori, symplectic rigidity
   
Type of project Fundamental research project
Research area Mathématiques
Method of financing FNS - Encouragement de projets (Div. I-III)
Status Completed
Start of project 1-4-2009
End of project 30-9-2010
Overall budget 167'980.00
Additional info http://p3.snf.ch/projects-125352#
Contact Félix Schlenk